This note announces three density theorems involving representations of Lie algebras and associative algebras. The first theorem describes the irreducible (possibly infinite dimensional) representations p of a Lie algebra g with an ideal ï such that the restriction of p to ï has some absolutely irreducible quotient representation. The second result is an embedding theorem for the irreducible representations of the Weyl algebras A niC over C (A n>c^C [t l9 • • • , t n , d/d^, • • • , 9/9^J, the

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... ssociative algebra of partial differential operators on n variables with coefficients in the polynomial ring C[t l9 • • • , t n ]). Our result is a sort of algebraic analogue of the uniqueness of the Heisenberg commutation relations, and has an application to irreducible representations of nilpotent Lie algebras via Dixmier's theory [5] . The third theorem describes the differentiably simple algebras having a maximal ideal. This result unifies the author's theorem [3] on differentiably simple rings with a minimal ideal, and Guillemin's theorem [7], [2] on the structure of a nonabelian minimal closed ideal of a linearly compact Lie algebra.